Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations

Area of Science:

• Mathematical modeling
• Epidemiology
• Control theory

Background:

• Opioid epidemics pose significant public health challenges.
• Dynamical models are used to understand epidemic spread.
• Controlling epidemic spread within safe boundaries is crucial.

Purpose of the Study:

• To develop a control strategy for a diffusion process model of the opioid epidemic.
• To prevent the epidemic model from exiting a specified bounded domain.
• To minimize the asymptotic exit rate of the controlled diffusion process.

Main Methods:

• Utilizing a compartmental model for opioid epidemic dynamics with random perturbations.
• Assuming degenerate diffusion and hypoellipticity of the diffusion operator.
• Deriving the Hamilton-Jacobi-Bellman equation for the optimal control problem.
• Relating the optimal control problem to a nonlinear eigenvalue problem.

Main Results:

• The Hamilton-Jacobi-Bellman equation was derived for the optimal control.
• The problem is closely linked to a nonlinear eigenvalue problem.
• A verification theorem was proven, offering sufficient conditions for optimality.

Conclusions:

• A mathematical framework was established for controlling opioid epidemic diffusion models.
• The derived Hamilton-Jacobi-Bellman equation provides a pathway to optimal control strategies.
• The findings contribute to the theoretical understanding of epidemic control in stochastic environments.